The Auto Company of Amenca (ACA) produces four types of cars: subcompact, compact, intermediate, and luxury. ACA also produces trucks and vans. Vendor capacities limit total production capacity to at most 1.2 million vehicles per year. Subcompacts and compacts are built together in a facility with a total annual capacity of 620,000 cars. Intermediate and luxury cars are produced in another facility with capacity of 400,000, and truck/van facility has a capacity of 275,000. ACA's marketing strategy requires that subcompacts and compacts must constitute at least half of the product mix for the four car types. The Corporate Average Fuel Economy (CAFE) standards in the Energy Policy and Conservation Act require an average fleet fuel economy of at least 27 mpg. Profit margins, market potential, and fuel efficiencies are summarized as follows: Type Profit Margin ($/vehicle) Market Potential (sales in '000) Fuel Economy (mpg) Subcompact Compact Intermediate Luxury Truck Van 150 225 250 500 400 200 600 400 300 225 325 100 40 34 15 12 20 25 a. The manager wants to determine the optimal production plan i.e., the optimal quantity of each vehicle type) that maximizes the total profit. Formulate the problem as a linear programming (LP) model and write down the mathematical formulation. (Hint: Four sets of constraints are needed: (1) capacity constraints at vendors and three production facilities, (2) the product mix requirement, (3) the requirement for the average fuel economy, and (4) the quantity of each vehicle type not exceeding the market potential.) (4 points] b. Solve your model in Excel and report the optimal production plan and the maximum profit. [3 points) Suppose that the marketing department may invest to increase the market potential for one of the vehicle types. Assuming that the profit margins remain unchanged, which vehicle type should ACA focus on? [1 point] d. Suppose that in addition to the existing truck/van facility with a capacity of 275,000 vehicles, an external manufacturer for trucks and vans is willing to offer an extra capacity of 30,000 vehicles at a total price of $6,000,000. Should ACA accept the offer? Please indicate if we can answer this question without resolving the models appropriately using the LP sensitivity report will be considered in grading. [2 points] C